The Nehari manifold for a semilinear elliptic equation involving a sublinear term

Abstract

The Nehari manifold for the equation $$ -\Delta u(x) = \lambda u(x) + b(x) \vert u(x)\vert^{\gamma - 2} u(x) $$ for $ x \in \Omega $ together with Dirichlet boundary conditions is investigated in the case where $ 1 < \gamma < 2$ . Exploiting the relationship between the Nehari manifold and fibrering maps (i.e., maps of the form $t \longrightarrow J(tu)$ where J is the Euler functional associated with the equation), we discuss how the Nehari manifold changes as $\lambda$ changes and show how this is linked to results on bifurcation from infinity which are associated with the problem.